Locally testable codes and PCPs of almost-linear length

@article{Goldreich2006LocallyTC,
  title={Locally testable codes and PCPs of almost-linear length},
  author={Oded Goldreich and Madhu Sudan},
  journal={Electron. Colloquium Comput. Complex.},
  year={2006}
}
  • Oded Goldreich, M. Sudan
  • Published 1 July 2006
  • Computer Science, Mathematics
  • Electron. Colloquium Comput. Complex.
We initiate a systematic study of locally testable codes; that is, error-correcting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject non-codewords with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we… 
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Locally testable codes and PCPs of almost-linear length
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It is shown that certain PCP systems can be modified to yield locally testable codes and PCPs of almost-linear length, and novel techniques in use include a random projection of certain codewords and PCP-oracles, an adaptation of PCP constructions to obtain "linear PCP -oracles" for proving conjunctions of linear conditions, and a direct construction of local testable (linear) codes of sub-exponential length.
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